Gas-Phase Heat of Formation Values for Buckminsterfullerene (C60

Aug 2, 2018 - ... a mean absolute deviation (MAD) and a root-mean-square deviation (RMSD) ... and aliphatic hydrocarbons and phenols, a MAD and a RMSD...
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Gas-Phase Heat of Formation Values for Buckminsterfullerene (C60), C70 Fullerene (C70), Corannulene, Coronene, Sumanene, and Other Polycyclic Aromatic Hydrocarbons Calculated Using Density Functional Theory (M06 2X) Coupled with a Versatile Inexpensive Group-Equivalent Approach Downloaded via UNIV OF CALIFORNIA SANTA BARBARA on August 2, 2018 at 20:37:50 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.

John A. Bumpus* Department of Chemistry and Biochemistry, University of Northern Iowa, Cedar Falls, Iowa 50614, United States S Supporting Information *

ABSTRACT: A straightforward procedure using density functional theory (M06 2X) coupled with a group-equivalent approach is described that was used to calculate gas-phase heat of formation (ΔfH°g,298) values for buckminsterfullerene (C60), C70 fullerene (C70), corannulene, coronene, and sumanene. This procedure was also used to calculate exceptionally accurate ΔfH°g,298 values for a variety of single-ring aromatic and 2−7 ring polycyclic aromatic hydrocarbons (PAHs) as well as a large selection of other hydrocarbons and phenols. The approach described herein is internally consistent, and results for C60, C70, corannulene, coronene, and sumanene are in very close agreement with results reported by others who used higher-level computational theory. Statistical analysis of a test set containing benzene and 18 two to seven ring PAHs demonstrated that by using this approach a mean absolute deviation (MAD) and a root-mean-square deviation (RMSD) of 0.8 and 1.3 kJ/mol, respectively, were achieved for reference/experimental ΔfH°g,298 values versus calculated/predicted ΔfH°g,298 values. For statistical analysis of a larger test set containing 235 aromatic and aliphatic hydrocarbons and phenols, a MAD and a RMSD of 1.2 and 1.9 kJ/mol, respectively, were achieved for reference/experimental ΔfH°g,298 values versus calculated/predicted ΔfH°g,298 values.

1. INTRODUCTION During the past several years, substantial progress has been made extending the use of a variety of computational approaches to calculate gas-phase heat of formation values (ΔfH°g,298) for C60 (buckminsterfullerene), C70 (C70 fullerene), corannulene, coronene, sumanene, and a variety of polycyclic aromatic hydrocarbons (PAHs).1−30 Karton and colleagues1,2 documented the use of very high level computational procedures (the Weizmann 1h (W1h) thermochemical protocol, double-hybrid density functional theory (DFT), second-order Møller−Plesset perturbation theory, spin-component-scaled MP2 (SCS-MP2) and G4MP2) to calculate ΔfH°g,298 values for C60, corannulene and sumanene. The article by Karton et al.1 was described as a tour de force by Dobek et al.,3 who demonstrated that a set of computationally less intensive theoretical procedures could be used to calculate values consistent with those of Karton et al.1 More recently, Chan et al.4 used another very high level computational approach (DSD-PBE-PBE/cc-pVQZ double-hydrid DFT) to calculate ΔfH°g,298 values for C60, C70, and larger fullerenes. The approaches used by Karton and his colleagues,1,2 Dobek et al.,3 and Chan et al.4 all require the use of several isodesmic or © XXXX American Chemical Society

other balanced reactions to arrive at the values reported. The present investigation describes how a different less intensive computational approach can be used to calculate accurate ΔfH°g,298 values. This approach yields results in very close agreement with those reported by Karton’s group,1,2 Dobek et al.,3 and Chan et al.4 Specifically, it was demonstrated that DFT using the M06 2X functional coupled with the 6-311+ +G(2df,2p)//6-311G* dual basis set and a straightforward group-equivalent approach can be used for the calculation of ΔfH°g,298 values for C60, C70, corannulene, coronene, and sumanene. This approach does not require the use of multiple isodesmic or other balanced reactions. It requires only the calculation of EElec (electronic energy) followed by calculation of H298 (enthalpy at 298.15 K) for the compounds being studied and access to accurate experimental ΔfH°g,298 values for compounds in the training set used to calculate groupequivalent values. Furthermore, this approach was shown to be readily applicable for calculating exceptionally accurate Received: April 8, 2018 Revised: June 16, 2018

A

DOI: 10.1021/acs.jpca.8b03321 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A ΔfH°g,298 values for a variety of single-ring aromatic and 2−7 ring PAHs as well as a large selection of other hydrocarbons and phenols

Table 1. Compounds Included in the Training Set and Their Experimental ΔfH°g,298 Valuesa compound methane propane n-heptane n-octane n-decane 2-methylpropane (isobutane) 3-methylpentane 2,2,4-trimethylpentane 2,2-dimethylpropane cubane benzene toluene phenol naphthalene phenanthrene pyrene anthracene propene trans-2-butene cis-2-butene isobutene 2-methyl-2-pentene 1,2-butadiene acetylene (ethyne) propyne cyclopentane methylcyclopentane 1.1-dimethylcyclopentane cyclohexane methylcyclohexane 1.1-dimethylcyclohexane cyclopentene 1-methylcyclopentene 1,2-dimethylcyclopentene cyclohexene 1-methylcyclohexene

2. COMPUTATIONAL METHODS 2.1. Overview. Quantum mechanics and atomic-, group-, or hybrid atomic/group-equivalent procedures have been used by several research groups8,10,11,13,18,23,31−34 to calculate ΔfH°g,298 values. The group-equivalent ΔfH°g,298 value for a given molecule can be calculated as follows Δf H °g,298 = H298 − ∑ nj ϵj

(1)

H298 is the enthalpy of the molecule at 298.15 K, nj is the number of atoms or groups present in the molecule, and ϵj is the group-equivalent value of group j. In this procedure, the electronic energy (EElec) was first determined for each compound using DFT, specifically, the M06 2X functional35,36 and the 6-311++G(2df,2p)//6-311G* dual basis set as implemented in the Spartan 10 or 14 (Wavefunction, Inc., Irvine, CA) suite of programs. Initial geometries were minimized using Hartree−Fock geometry minimization and the 6-31G* basis set. For correction of EElec to E298, requisite parameters, unscaled zero-point energies (ZPE), and thermal correction (HT) values) for all compounds were determined following calculation of equilibrium geometries and frequency calculations using the M06 2X functional and the 6-31G* basis set. Finally, RT was added, converting E298 to H298. Frequency calculations yielded no imaginary frequencies. Calculations were performed on a Dell Optiplex 980 desktop computer equipped with an Intel(R) Core i7 CPU, a 64-bit operating system, 8.0 GB of RAM, and a 1 TB hard drive or on a Dell Latitude laptop computer equipped with an Intel(R) Core i7 CPU, a 64-bit operating system, 8.0 GB of RAM, and a 500 GB hard drive. Following calculation of H298, atom- or group-equivalent values were determined by least-squares regression fit of eq 1 to molecules in the training set. Least-squares regression was accomplished using Excel’s data analysis tool for regression under the Data menu. Compounds included in the training set are presented in Table 1, and group-equivalent values calculated using this training set are presented in Table 2. Calculated EElec and H298 values for all compounds in this investigation are presented in Table S4. A technical limitation of Excel’s regression analysis program is that it restricts the number of dependent variables to 16. This was problematic because 34 group values were required for this investigation. To address this issue, ΔfH°g,298 values for a training set of 13 compounds (methane, propane, n-heptane, n-octane, n-decane, 2-methylpropane, 3-methylpentane, 2,2,4-trimethylpentane, 2,2-dimethylpropane, cubane, benzene, toluene, and phenol) were used initially to calculate group-equivalent values for methane (CH4) and the methyl (−CH3), methylene bridge (a.k.a. methylene spacer or methanediyl) (−CH2−), and methine (>CH−) groups and carbon (>CCH− >C< Ar CHa (or Ca-ring (H)) Ar C no H Ar C−CH3 Ar C−OH phenyl (−C6H5) −OH −CHCH2 −HCCH− (trans) −HCCH− (cis) H2CC< >CCH− HCCCH2 CH C− (no H) −CH2− in a 5-membered ring >CH− in a 5-membered ring >C< in a 5-membered ring −HCCH− in a 5-membered ring HCC< in a 5-membered ring >CC< in 5-membered ring −CH2− in a 6-membered ring >CH− in a 6-membered ring >C< in a 6-membered ring −HCCH− in a 6-membered ring HCC< in a 6-membered ring Ca-fused a-ring (Ca-ring, Ca-ring, Ca-fused a-ring)b Ca-fused a-ring (Ca-fused a-ring Cafused a-ring Ca-fused a-ring)b Ca-fused a-ring (Ca-ring, Ca-fused a-ring, Ca-fused a-ring)b Ca-ring(H), (Ca-fused, Ca-fused)d a

C−(C)−(H)3 −C−(C)2−(H)2 −C−(C)3−(H) −C−(C)4 CB−(H) CB−(C)

Ct−(H) Ct−(C)

CBF−(CB)2(CBF)c

group-equivalent value (hartrees) −40.422992 −39.844747 −39.267306 −38.688582 −38.111306 −38.689629 −38.111443 −77.956190 −113.845848 −231.559587 −75.734406 −77.955808 −77.378421 −77.378029 −77.378092 −76.798135 −116.070136 −38.688739 −38.113037 −39.266847 −38.689274 −38.111255 −77.378244 −76.799848 −76.221806 −39.267119 −38.688677 −38.111243 −77.377970 −76.799580 −38.111633

CBF−(CBF)3c

−38.110052

CBF−(CB)(CBF)2c

−38.110374

CB−H−(CBF)2d

−38.687934

b

Ar C = aromatic carbon. Abbreviations from NIST WebBook’s group additivity applet (http://webbook.nist.gov/chemistry/grp-add/ ga-app/). cFor abbreviations, see ref 27 and Table S3. dSee text for explanation and Table S3.

mutually supportive. For norbornane, the difference in ΔfH°g,298 values is only 0.2 kJ/mol. Thus, it would be reasonable to simply calculate and use the average value. For compounds in which the difference is substantial, it would be appropriate to calculate heat of formation values using other procedures (isodesmic reactions, T1, Gaussian G2, G3, G4, G3MP2, G4MP2, etc.) to determine which values (if any) are mutually supportive. Structural group assignments are presented in Table S5. In addition to values already mentioned, group-equivalent values were also calculated for carbons in PAHs present in different environments, as described by Stein et al.27 A graphic description of the abbreviations/designations used by Stein et al.27 is presented in Table S3. In addition to these abbreviations/designations, a group-equivalent value was calculated for aromatic carbons in acenes (e.g., anthracene, naphthacene, pentacene, hexacene, etc.) bonded to a hydrogen and two fused aromatic ring carbons. This group is designated C

DOI: 10.1021/acs.jpca.8b03321 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A coronene, an experimental ΔfH°g,298 value of 279.7 ± 7.1 kJ/ mol was used. This value was determined by adding the heat of sublimation value (ΔsubH°) for coronene determined by Torres et al.57 to its solid phase heat of formation (ΔfH°s) value determined by Nagano.58 For perylene, the ΔsubH° value determined by Nass et al.,59 was added to the ΔfH°s value reported by Cox and Pilcher49 to give the ΔfH°g,298 value of 306.0 kJ/mol that was used in this investigation. This same value was recommended by Slayden and Liebman17 and used by Blanquart and Pitsch22 in their investigation. The ΔfH°g,298 values for two methyl-substituted polycyclic hydrocarbons (1methylnaphthalene and 2-methylnaphthalene) were also included in this test set.60 For benzene, the value listed in the ATcT (version 1.118, ATcT.anl.gov) was used. Regarding the ΔfH°g,298 value used for naphthacene, experimental values reported for this PAH vary considerably. Karton and Martin61 addressed this issue using high-level (W1F12) computational theory, calculating a ΔfH°g,298 value of 309.03 kJ/mol. In their investigation, this same approach was used to calculate ΔfH°g,298 values for two other acenes, naphthalene and anthracene; both calculated values were slightly less than accepted experimental ΔfH°g,298 values. Karton and Martin61 reasoned that their calculated ΔfH°g,298 value for naphthacene might also be slightly underestimated. In further investigations involving the following isodesmic reaction

Dobek et al.,3 Chan et al.,4 Yu et al.,18 Sun et al.,21 and Karton and his colleagues.1,2 It should be noted that calculated results for all compounds containing one or more carbons bonded only to other fused carbons (CBF(CBF)3) are uniquely dependent upon the experimental ΔfH°g,298 value (224.0 ± 2.2 kJ/mol) for pyrene used in the training set. As this study progressed, it appeared that this value might be too low by a few tenths of a kJ/mol. This is not problematic for the compounds in the test sets that have only 1−6 such carbons in their structures. However, for C60 and C70, any small error is multiplied 60 and 70 times, respectively. This issue was addressed as follows: It was assumed that several reported1−4,18,21 ΔfH°g,298 values for C60 and C70 were equally accurate, within limits of reported uncertainty. These values were then used to calculate ΔfH°g,298 values for pyrene by substituting them for pyrene in the training set. Then, group-equivalent values were calculated by linear regression analysis as described. These group values were subsequently used to calculate a composite ΔfH°g,298 value for pyrene of 224.4 ± 0.5 kJ/mol. Ultimately, a composite groupequivalent value was calculated and used. Calculated ΔfH°g,298 values for all compounds in the test sets containing carbons bonded only to other fused carbons are reported using groupequivalent values derived from both the experimental ΔfH°g,298 value and the composite ΔfH°g,298 value. Data used for calculation of the composite ΔfH°g,298 value and its groupequivalent value are presented in Table S6. A comment regarding the ΔfH°g,298 value used for anthracene in the training set is warranted here. Roux et al.’s42 recommended value is 229.4 ± 2.9 kJ/mol. However, as noted above, a ΔfH°g,298 value of 227.1 ± 5.6 kJ/mol was used. This is an experimental value, rather than an average of experimental values, and it is within the range of uncertainty for the recommended value. Anthracene is important because it is required for calculation of the group value for a carbon bonded to a hydrogen and two fused carbons in acenes. This group value was used to calculate the ΔfH°g,298 value for naphthacene, which is consistent with the reference ΔfH°g,298 value for this compound. As this study progressed, it also became apparent that this approach would prove to be exceptionally accurate for calculation of ΔfH°g,298 values for a variety of other aromatic and nonaromatic hydrocarbons and phenols. Thus, in addition to the test set (Test Set 1) used to compare this study with that of Allison and Burgess,23 another test set (Test Set 2) was developed and used to assess the usefulness of this approach to calculate/predict ΔfH°g,298 values for a variety of alkanes, alkenes, alkynes, cycloalkanes, cycloalkenes, phenols, singlering aromatic compounds, and several PAHs. The experimental ΔfH°g,298 values used for several PAHs in the test set were those that had been critically evaluated by Roux et al.42 for which there is general agreement with regard to their accuracy. Other ΔfH°g,298 values for compounds included in Test Set 2 were among those listed in the ATcT Tables, the NIST WebBook,47 Cox and Pilcher’s Thermochemistry of Organic and Organometallic Compounds,49 or Pedley et al.’s Thermochemical Data of Organic Compounds50 and/or from references in the literature.51−118 A detailed list of literature references for reference/experimental ΔfH°g,298 values selected for use in the training set and the two test sets is provided in Table S1. 2.3. Selection of the Computational Approach. Several investigations have successfully used an approach in which ΔfH°g,298 values were calculated using atom-, group-, or hybrid

2 anthracene → naphthalene + naphthacene

they calculated a ΔfH°g,298 value of 312.13 kJ/mol. According to Karton and Martin,61 the average (310.67 ± 4.184 kJ/mol) of this this value and their previously calculated value (309.03 kJ/mol) represents “a more realistic estimate”.61 This average value was used as the reference value for naphthacene in the test sets. Like naphthacene, reported experimental ΔfH°g,298 values for benz[a]anthracene are suspect. Thus, for benz[a]anthracene, the reference ΔfH°g,298 value used in the test sets was determined as follows. Consider that, within their ranges of uncertainty, ΔfH°g,298 values for naphthalene (150.6 kJ/mol ±1.5) and anthracene (227.1 ± 5.6 kJ/mol) are accurate. The difference between these two ΔfH°g,298 values is 76.5 kJ/mol and represents the same group (H−CC−CC−H) value that would be added to phenanthrene (ΔfH°g,298 = 202.2 kJ/ mol) in a theoretical reaction resulting in synthesis of benz[a]anthracene. Thus, group theory gives a mean ΔfH°g,298 value of 278.7 kJ/mol used in the test sets for benz[a]anthracene. It should be emphasized that the ΔfH°g,298 value for naphthacene is based solely on high-level computational theory and the ΔfH°g,298 value for benz[a]anthracene is based on experiment coupled with group theory. Neither is based solely on experiment. Nevertheless, they are included in the test set so that there are credible reference values for all 19 values. For this reason, values in the test sets are referred to as reference/experimental values. Because the ΔfH°g,298 values for C60 and C70 are still under scrutiny, they were not included in the test set. And because reliable experimental ΔfH°g,298 values for corannulene and sumanene are not available, these values could not be included in the test set. However, computed ΔfH°g,298 values for all four of these compounds as well as the computed and experimental ΔfH°g,298 values for coronene were compared in detail with values computed by several other research groups, including D

DOI: 10.1021/acs.jpca.8b03321 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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Table 3. Comparison of ΔfH°g,298 Values in Test Set 1 Calculated Using the DFT/Group-Equivalent Approach (This Study) with Reference/Experimental Values and Values Calculated in the Computational Study of Allison and Burgess23 compound 1-methylnaphthalene 2-methylnaphthalene acenaphthene acenaphthylene anthracene benz[a]anthracene benzene biphenyl biphenylene chrysene coronene fluoranthene fluorene naphthacene naphthalene perylene phenanthrene pyrene triphenylene MAD RMSD

reference/experiment

this study

exp−calc this study

Allison and Burgess23

exp−calc Allison and Burgess23

ΔfH°g,298 (kJ/mol) 116.9 116.1 156.8 265.5 227.1 278.7 83.2 180.3 420.4 263.5 279.7 289.8 179.4 310.7 150.6 306.0 202.2 224.0 270.1

ΔfH°g,298 (kJ/mol) 117.2 116.3 156.5 265.2 227.1 279.7 83.2 179.5 419.2 261.9 280.1 287.0 182.9 311.7 150.6 304.1 202.2 224.0 269.7

(kJ/mol) −0.3 −0.2 0.3 0.3 0.0 −1.0 0.0 0.8 1.2 1.6 −0.4 2.8 −3.5 −1.0 0.0 1.9 0.0 0.0 0.4 0.8 (1.1)a 1.3 (1.5)a

ΔfH°g,298 (kJ/mol) 113.5 108.7 150.7 259.8 222.6 277.1 75.2 174.2 410.9 271.1 296.7 277.9 179.6 310.5 141 319.2 202.7 221.3 275.1

(kJ/mol) 3.4 7.4 6.1 5.7 4.5 1.6 8 6.1 9.5 −7.6 −17 11.9 −0.2 0.2 9.6 −13.2 −0.5 2.7 −5 6.3 7.7

MAD and RMSD values calculated omitting ΔfH°g,298 values for those compounds (underlined and in bold) included in the training set.

a

the root-mean-square deviation (RMSD) were calculated as follows

atom/group-equivalent energies determined using quantum mechanical calculations.8,10,11,13,18,23,31−34 The approach described here was inspired by Wiberg31 who used ab initio theory to calculate group values and by Mole et al.32 and Rice and her colleagues33,34 who used DFT to calculate atom-, group-, or hybrid atom/group-equivalent values. The DFT/group-equivalent approach described here differs from those used by other investigators32−34 in several ways. First of all, the M06 2X functional was used. The M06 2X functional is one of the Minnesota functionals developed by Truhlar’s group.35,36 It has been specifically recommended for calculation of thermochemical parameters and is conveniently included in several quantum mechanical packages (e.g., QChem, Gaussian, Gamess, Spartan). Mole et al.32 used atomequivalents, whereas Rice’s group33,34 used both atom- and group-equivalents. The procedures described here to calculate and use group-equivalents (Table 2) differ from those used by Rice and her colleagues.33,34 Rice et al.33 calculated and used atom-equivalents for single-bonded C, H, N, and O atoms and a second set of atom-equivalents using C, H, N, and O atoms involved in multiple-bond environments. This second category included C, H, N, and O atoms in aromatic compounds. Because their investigations were developed primarily for energetic nitrogen-containing compounds, Byrd and Rice34 also calculated group-equivalent values for −NO2 groups bound to nitrogen, oxygen, singly bonded carbons, and carbons in multiple-bonded environments. A group-equivalent for carbon bonded to an azide group was also calculated. Still another difference between the DFT/group-equivalent approach described here and that of Mole et al.32 and Rice and her colleagues33,34 is that here H298, the enthalpy of the molecule at 298.15 K, is used (see eq 1) instead of EElec. 2.4. Statistical Analysis and Calculation of Error. For data in the test sets, the mean absolute deviation (MAD) and

MAD = RMDS =

1 n

∑ |Δf H °g,298(expt) − Δf H °g,298(calc)| 1 n

∑ (Δf H °g,298(expt) − Δf H °g,298(calc))2

It should be noted that the 95% confidence interval is roughly approximated by doubling the RMSD.119,120 For statistical analyses, all experimental and calculated ΔfH°g,298 values were rounded to the nearest 0.1 kJ/mol. For comparisons between calculated values for C60 and C70, corannulene, and sumanene, it was important to provide an approximation of the error or uncertainty for their respective calculated ΔfH°g,298 values. This approximation was accomplished as follows. We started with the fact that the NIST WebBook states that for the ΔfH°g,298 values for PAHs recommended by Roux et al.42 “the evaluated uncertainty limits are on the order of 2−4 kJ/mol”. Because calculation of ΔfH°g,298 values for C60 and C70 rely exclusively on the accuracy of the reported experimental ΔfH°g,298 value (and error) for pyrene, it was reasonable to estimate uncertainty by calculating the error per nucleon in pyrene. For this, an uncertainty value of 4 kJ/mol was used to calculate an uncertainty of 0.020 kJ/mol/nucleon. This value was then multiplied by the number of nucleons in C60 and C70. In this manner, error values for the ΔfH°g,298 of C60 and C70 were estimated to be ±14.4 and ±16.8 kJ/mol, respectively. A slightly different approach was used to estimate error associated with calculated ΔfH°g,298 values for corannulene and sumanene. Four PAHs (anthracene, naphthalene, phenanthrene, and pyrene) were used in the training set to calculate group values for these compounds. Of these, the reported error associated with accepted experimental ΔfH°g,298 values was E

DOI: 10.1021/acs.jpca.8b03321 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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Table 4. Comparison of ΔfH°g,298 Values Calculated for Compounds in Test Set 1 Using the DFT/Group-Equivalent Approach (This Study) with Values Calculated Using Several Other Approaches reference this study Allison and Burgess23 Blanquart and Pitsch22 Sivaramakrishnan et al.20 Yu et al.18 Yu et al.18 Rayne and Forest26 Welsh et al.14 Alberty and Reif5,6 Wang and Frenklach12 Wang and Frenklach12 Wang and Frenklach12 Wang and Frenklach12 Herndon13 Herndon13 Herndon13 Schulman et al.8 Schulman et al.8 Schulman et al.8 Peck et al.10 Disch et al.11 Peck et al.,10 Disch et al.11 Zauer29

MAD (kJ/mol) 0.8 6.3 3.0 9.2 8.1 9.7 16.5 31.7 7.3 50.3 10.4 4.1 7.2 7.6 7.7 7.1 10.4 8.4 5.4 5.9 5.2 4.8 17.5

(1.1)a (1.6)b (7.9)b (5.5)b (6.6)b (18.5)c

(9.3)b (7.4)b (4.1)b

(9.7)b

RMSD (kJ/mol) 1.3 7.7 4.9 11.5 12.2 16.0 17.3 64.8 10.5 57.6 17.3 5.3 9.0 9.9 10.1 9.3 14.9 11.5 7.8 8.4 8.1 7.2 35.5

(1.5)a (2.1)b (9.3)b (7.7)b (9.6)b (31.7)c

(13.9)b (10.4)b (5.2)b

(11.6)b

n

comment

19 (14)a 19 8 (7)b 16 (15)b (13) (12)b 13 (12)b 13 16 (15)c 8 11 11 11 11 14 14 14 13 (12)b 13 (12)b 13 (12)b 13 6 13 16 (15)b

DFT/group-equivalent DFT/group-equivalent/group correction mixed G3MP2//B3 and group correction DFT ring conserved isodesmic reactions DFT homodesmic bond-centered group additivity G4MP2 comparative molecular field analysis group additivity semiempirical (AM1) semiempirical (AM1 GC2) semiempirical (AM1 GC4a) semiempirical (AM1 GC4b) HF ab initio and regression analysis HF ab initio and regression analysis HF ab initio and regression analysis HF STO-3G/group-equivalent HF 3-21G/group-equivalent HF 6-31G*/group-equivalent HF 6-31G*/group-equivalent HF 6-31G*/group-equivalent HF 6-31G*/group-equivalent (combined) semiempirical (AM1 corrected)

Values in parentheses represent analyses in which ΔfH°g,298 values of compounds in the training set have been omitted. bValues in parentheses represent analyses in which the ΔfH°g,298 value for coronene was omitted. cValue in parentheses represent analyses in which the ΔfH°g,298 value for biphenylene was omitted.

a

greatest for anthracene (ΔfH°g,298 = 227.1 ± 5.6 kJ/mol). Thus, the error per nucleon was calculated for anthracene (0.031 kJ/nucleon) and then multiplied by the number of nucleons in corannulene and sumanene to provide estimated error values of ±7.8 and ±8.2 kJ/mol, respectively. It is noted that this treatment is the same as dividing the error value of pyrene or anthracene by their respective molecular weights and then simply multiplying by the molecular weight of the compound in question.

and described by Allison and Burgess.23 This is of interest because the procedures developed by Allison and Burgess23 were selected and used to calculate/predict ΔfH°g,298 values for PAHs listed in the NIST Polycyclic Aromatic Hydrocarbon Structure Index (http://pah.nist.gov/). This very important and useful database provides (among many other things) reasonably accurate calculated/predicted ΔfH°g,298 values for PAHs in those instances in which accurate experimental ΔfH°g,298 values are not yet available. In addition to the study by Allison and Burgess,23 the relative accuracy of the DFT/group-equivalent approach described here was compared with results of several studies in which a variety of other computational approaches were used to predict/calculate ΔfH°g,298 values for PAHs (Tables 4 and S7). Comparison of the results of Allison and Burgess23 with those reported here using the DFT/group-equivalent approach (Table 3) is straightforward because computed and reference/experimental ΔfH°g,298 values are available in both studies for all 19 compounds in the test set. For studies conducted by other research groups (Tables 4 and S7), the number of computed ΔfH°g,298 values reported ranged from 6 to 16. It is, therefore, reasonable to view with some caution direct comparisons of data using the DFT/group-equivalent approach reported here with data from these other studies. Nevertheless, it does appear that the DFT/group-equivalent approach is the most accurate approach of those summarized in Tables 4 and S7. Of the results in the 14 publications compared, only the approaches described by Blanquart and Pitsch,22 Schulman et al.,8 and Wang and Frenklach12 achieved a MAD value less than 4.184 kJ/mol. Using a multilevel ab initio approach (the mixed (G3(MP2)//B3) method), Blanquart and Pitsch22 first

3. RESULTS AND DISCUSSION 3.1. Accuracy of the DFT/Group-Equivalent Approach to Predict ΔfH°g,298 Values for C60, C70, Polycyclic Aromatic Hydrocarbons, other Aromatic Compounds, Aliphatic Compounds, and Phenols. Results (Table 3) for Test Set 1 consisting of benzene and 18 PAHs demonstrate that all of the ΔfH°g,298 values calculated using the DFT/groupequivalent approach were within ±4.184 kJ/mol (±1.0 kcal/ mol) of reference/experimental values. Further statistical analysis revealed that a MAD of 0.8 kJ/mol and a RMSD of 1.3 kJ/mol/mol were achieved. When compounds used in the training set were omitted from the test set, MAD and RMSD values of 1.1 and 1.5 kJ/mol, respectively, were calculated. This test set was selected so that a direct comparison could be made with the study by Allison and Burgess23 for which MAD and RMSD values of 6.3 and 7.7 kJ/mol, respectively, were calculated (Table 3). Further analysis showed that 31.6% (6/ 19) of the calculated ΔfH°g,298 values were within ±4.184 kJ/ mol (±1.0 kcal/mol) of the reference/experimental values and 73.7% (14/19) were within ±8.368 kJ/mol (±2.0 kcal/mol). Results show clearly that the DFT/group-equivalent approach used here is substantially more accurate than that developed F

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Table 5. Computed Gas Phase Heat of Formation (ΔfH°g,298) Values for Buckminsterfullerene (C60 Fullerene), C70 Fullerene, Corannulene, Coronene, and Sumanene reference

buckminsterfullerene (C60) (kJ/mol)

C70 fullerene (kJ/mol)

Schulman et al.8

corannulene (kJ/mol) 490.4 504.2 516.3 488.7 516.3

Peck et al.10 Disch et al.11

Steele et al.121 Schulman and Disch15

2656 ± 25

Cioslowski et al.16

2586.1 2580.7 2509.6 2529.6 2331.3 2511.7 1865.6 3084.4

2751.8 2771.1 2743.9 2789.9

2589.1

2686.1

Yu et al.18 Grimme19

Sivaramakrishnan et al.20 Sun et al.21 Blanquart and Pitsch22 Karton et al.1 Dobek et al.3 Allison and Burgess23 Chan et al.4 Wan and Karton2 this work this work (composite approach)

coronene (kJ/mol)

sumanene (kJ/mol)

256.1 259.4 258.2 256.5 257.3 256.5 277.4 283.7 288.7 277.4 283.7 290.4 292.0 293.3 300.8 300.4 274.1

491.2 504.2

314.6 326.8

502.9

308.4 292.43

2521.6 ± 13.6 2531 ± 15.0

485.2 ± 7.9 484.0 ± 4.0 498.1

2520.0 ± 20.7 2511.7

2683.4 ± 17.7

2508.3 ± 14.4 2520.4 ± 15.1

2681.5 ± 16.8 2695.6 ± 17.6

calculated preliminary ΔfH°g,298 values followed by use of group correction factors to ultimately achieve the accurate ΔfH°g,298 values reported. MAD and RMSD values of 3.0 and 4.9 kJ/mol were calculated for their test set of eight PAHs. However, when coronene was removed from the test set, MAD and RMSD values of 1.6 and 2.1 kJ/mol were calculated. The computational approach used by Blanquart and Pitsch22 may be described as relatively high level computational theory. Thus, it is not surprising that application of such a high level of theory would result in the accurate predictions/calculations observed. MAD and RMSD values of 4.1 and 5.2 kJ/mol, respectively, were calculated using data from Schulman et al.,8 (ab initio (HF 6-31G*)/group-equivalent approach). However, these values were only achieved when the calculated and experimental ΔfH°g,298 values for coronene were omitted from their test set of 13 PAHs. Surprisingly, the semiempirical approach used by Wang and Frenklach12 was also found to be rather accurate. Semiempirical procedures are useful because they are very rapid and provide, in many cases, good initial geometries and rough

486.5 ± 7.8 487.5 ± 1.3

296.7

280.1 ± 9.3 281.3 ± 1.5

530.3 535.3 527.1 528.3

± ± ± ±

8.0 9.0 8.2 1.5

estimates of a compound’s thermochemical and other properties. They are typically not used for calculation of exceptionally accurate ΔfH°g,298 values. Indeed, Wang and Frenklach’s12 initial application of unmodified AM1 theory resulted in clearly unacceptable MAD and RMSD values of 50.3 and 57.6 kJ/mol, respectively. Nevertheless, after calculation and application of group correction factors, their AM1 GC4a approach resulted in credible MAD and RMSD values of 4.1 and 5.3 kJ/mol, respectively, for their test set of 11 PAHs. The ΔfH°g,298 values for C60, C70, corannulene, coronene, and sumanene calculated/predicted using the protocol described herein are presented in Table 5 along with values calculated/predicted by other investigators. Experimental values for C60 and C70121−137 are presented in Table 6. The accuracy of calculated and experimental ΔfH°g,298 values for these compounds is discussed individually and in detail later in the Results and Discussion section. Although it was not the primary objective of this investigation, it became apparent that the DFT/groupequivalent approach is also useful for calculating very accurate G

DOI: 10.1021/acs.jpca.8b03321 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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Table 6. Experimentally Determined ΔfH°s and ΔfH°g,298 Values for Buckminsterfullerene (C60 Fullerene) and C70 Fullerene reference

buckminsterfullerene (C60 fullerene) ΔfH°s (kJ/mol)

Steele et al. (1992)121 Diogo and Minas da Pledade (1993)124 Beckhaus et al. (1994)126 Kiyobayashi and Sakiyama (1993)125 Beckhaus et al. (1992)123 Kolesov et al. (1996)127 Xu-wu et al. (1996)130 Xu-wu et al. (1998)131 Rojas-Aguilar (2002)128 Diky and Kabo (2000)136 Rojas-Aguilar and Martinez-Herrera (2005)135 Rojas-Aguilar and Martinez-Herrera (2005)135 mean ± SD reference

2422 ± 14.0 2278.1 ± 14.4 2327 ± 17.0 2273 ± 15.0 2280.2 ± 5.6 2355 ± 15.0 2359.6 ± 9.7 2336 ± 20.0 2288.5 ± 29.5 2346 ± 12.0 2277.7 ± 27.4 2293.1 ± 44.8 2319.7 ± 46.0 C70 fullerene ΔfH°s (kJ/mol)

Beckhaus et al. (1994)126 Pimenova et al. (1997)129 Kiyobayashi and Sakiyama (1993)125 Xu-wu et al. (1998)131 Diogo et al. (1997)132 Pimenova et al. (2003)133 Rojas-Aguilar (2004)134 Diky andKabo (2000)136 Rojas-Aguilar and Martinez-Herrera (2005)135 Rojas-Aguilar and Martinez-Herrera (2005)135 mean ± SD

2555 ± 12.0 2439 ± 37.0 2375 ± 18.0 2407 ± 22.0 2577.8 ± 16.2 2452 ± 33.0 2554 ± 57.6 2555 ± 22.0 2558 ± 29.0 2537 ± 37.0 2501.0 ± 74.6

buckminsterfullerene (C60 fullerene)a ΔfH°g,298 (kJ/mol) 2608.6 ± 15.2 2464.6 ± 15.6 2513.3 ± 18.2 2459.3 ± 16.2 2466.5 ± 6.8 2541.3 ± 16.2 2545.9 ± 10.9 2522.3 ± 21.1 2474.8 ± 30.7 2532.3 ± 13.2 2464.0 ± 28.6 2479.4 ± 46.0 2506.0 ± 46.0 C70 fullereneb ΔfH°g,298 (kJ/mol) 2749.5 2633.5 2569.5 2601.5 2772.5 2646.5 2748.5 2749.5 2752.5 2731.5 2695.5

± ± ± ± ± ± ± ± ± ± ±

13.7 38.7 19.7 23.7 17.9 34.7 59.3 23.7 30.7 38.7 74.6

Calculated with the ΔsubH° value of 186.3 ± 1.2 kJ/for C60 determined by Martinez-Herrera et al. (2015).137 bCalculated with the ΔsubH° value of 194.5 ± 1.7 kJ/mol for C70 determined by Martinez-Herrera et al. (2015).137 a

ΔfH°g,298 values for one-ring aromatic hydrocarbons as well as 2−7 ring PAHs, aliphatic hydrocarbons, and phenols. This observation was confirmed by the results presented in Table 7. In Table 7, the calculated ΔfH°g,298 values for 235 compounds in Test Set 2 are compared with reference/experimental ΔfH°g,298 values. Initial results showed that this approach is, indeed, very accurate. Nevertheless, initial results also demonstrated that ΔfH°g,298 values for cyclopentane and cyclohexane derivatives were consistently overestimated by about 4 and 2 kJ/mol, respectively. For this reason, groupequivalent values were specifically determined and used to calculate ΔfH°g,298 values for compounds having carboncontaining groups (−CH2−, >CH, and >C